International Journal of Fluid Machinery and Systems DOI: http://dx.doi.org/10.5293/IJFMS.2020.13.2.476 Vol. 13, No. 2, April-June 2020 ISSN (Online): 1882-9554 Original Paper Surface Mesh Generation in Parametric Space Using a Riemannian Surface Definition Cui Dai1, Zhaoxue Wang 1, Liang Dong2, Yiping Chen1, Junfeng Qiu1 1School of Energy & Power Engineering, Jiangsu University, Zhenjiang 212013, Jiangsu, China, [email protected], [email protected], [email protected], [email protected] 2Research Center of Fluid Machinery Engineering and Technology, Jiangsu University, Zhenjiang 212013, Jiangsu, China, [email protected] Abstract In order to solve the problem of generating distortion elements in the mapping from parameter space to real space, and the boundary coincidence of the mesh generated by the software quality, an approach for parametric surface mesh generation based on Riemannian metric, combined with Delaunay triangulation and AFT is proposed. In our algorithm, the boundary curves are discretized based on the proximity and curvature of the curves in the model after derivation the correlation of curve length between parametric space and real space. Background meshes of parametric space were generated by using improved AFT, and could improve the efficient of the algorithm and control element sizing and metric values. When background mesh of parametric space were refined, to counteract mapping distortion, the traditional Delaunay incremental insertion kernel is replaced by inserting the center of triangle circumscribed ellipse, and the algorithm for locating ellipse center and judging whether nodes within ellipse. In this paper, the details of the surface mesh generated by the algorithm are introduced in detail. The algorithm proposed in this paper has the characteristics of reliable algorithm, high mesh generation efficiency and mesh quality. Finally, the reliability of the proposed algorithm is verified by an example of surface mesh generation. Key words: Surface mesh generation; Riemannian metric; advancing front method; Delaunay triangulation; Background mesh 1. Introduction Surface mesh generation is one of the most difficult and yet important prerequisites of mesh generation in three dimensions. The surface mesh influences the volume mesh close to the boundary of the domain. The smoothness of the surface mesh, and the conformity of this mesh to the domain boundary, directly affects the quality of any further finite element simulation based on the mesh. The accuracy of the numerical solutions and the convergence of the computation scheme are strongly related to the quality of the underlying surface meshes [1] (Since surface meshes define external Received March 3 2020; accepted for publication April 9 2020: Review conducted by Xuelin Tang. (Paper number O20005k) Corresponding author: Dong Liang, [email protected] 476 and internal boundaries of computational domains where boundary conditions are imposed). There are two popular surface mesh generation methods [2]: direct method and indirect method. The indirect method is conceptually straightforward as a two-dimensional mesh is generated in the parametric domain and thus it is expected to be faster than the direct approach. In addition, Indirect method is strong the adaptiveness for complex curved-surface, so indirect method are among the most commonly used surface mesh generation. The problem with the methods is the generation of a mesh which conforms to the metric of the surface. In fact, they aimed to minimize the error in the polyhedral approximation of the surface indirectly in the parametric space without paying attention to the quality of the resulting mesh. The mesh in the parametric surface is usually anisotropic, due to the metric deformation from the surface to its parametric domain. Thus, for people in finite element computation, the problem is reduced to the generation of an anisotropic mesh in the parametric domain. Cuillière[3] developed automatic mesh generation procedures based on advancing front methods (AFT) and featuring a priori nodal density calculations, and focused on the discretization of 3D parametric surfaces with strong variations of curvature with respect to a nodal density function with steep gradients. This method can be more accurately calculate element distortion, but may not be so general. In [4], Delaunay method was extended to anisotropic context of 2D domains, and a Riemannian metric map is introduced to remedy the mapping distortion from object space to parametric domain. The algorithm is easier to implemented and efficient, but it doesn’t suit to the surface that differs very much from interior to the boundary. In [5], Guan presented a new mesh generation procedure which is suggested for the triangulation of general combined parametric surfaces using an advancing front approach and metric tensor. The calculation and interpolation method of arbitrary points in surface’s parametric space are detailed. Tristano [6] presented a method for meshing 3D parametric surfaces using the advancing front method with a Riemannian surface definition. The details of the creation of the metric map used to determine the amount of distortion of the elements in parametric space are given along with the details of an advancing front algorithm that utilizes the metric map. The method overcomes anomalies found in the warped parametric space and direct 3D methods. Borouchaki[7]presented an indirect method for meshing parametric surfaces conforming to a user-special size map. First, from the size specification, a Riemannian metric is defined so that the desired mesh is one with unit length edges with respect to the related Riemannian space (the so-called ‘unit mesh’). Then, based on the intrinsic properties of the surface, the Riemannian structure is induced into the parametric space. Finally, a unit mesh is generated completely inside the parametric space so that it conforms to the metric of the induced Riemannian structure. Experiments [8] prove that Riemannian metric has well effect on eliminating mapping distortion. If only Riemannian metric is reasonably defined, the quality of surface mesh in 2D parametric domain can be controlled. A significant amount of research has gone into meshing surfaces using a metric map with Delaunay triangulation or Advancing Front Technique independently. However, those mesh generation methods have their own disadvantages (for example, AFT cannot assure the quality of the generated mesh to meet the standards, Delaunay triangulation needs to look for the triangle with the biggest dimensionless radius in generating internal nodes, so can be time-consuming. In addition, they calculate the size and metric of the surface by calling related functions in the surface mesh generating process and tend to be slower and less robust. Little attention has been paid to the automatic generation of mesh size and metric functions on a background mesh simultaneously. Most this work is only devoted to the automatic generation of mesh size functions [9-11]. In order to strikes a balance between surface mesh quality and computational cost, this paper proposes an approach for parametric surface mesh generation based on Riemannian metric, combined with Delaunay triangulation and AFT. The control element sizing and metric values are provided in a set of discrete points (vertices of a background triangular mesh) covering the parametric domain. Those discrete points are generated by AFT. The interpolation on individual triangles of background control mesh is then used to approximate the relevant mesh control function in the rest of the parametric space. Delaunay triangulation is then used to refine the mesh of parametric domain. 477 2. Overall Algorithm In this section, the overall surface mesh generation scheme is presented. Algorithmic details for each of the major steps in this scheme will be presented later in the paper. More specifically, the proposed algorithm involves the following steps: Read geometry information on parametric surfaces, and determine the corresponding boundary curves. According to the boundary curves on discrete curves with given metrics, make out the coordinate of all discrete points（section 3.1）. According to the discrete results of parameter curves, generate background mesh by using improved AFT and store the metrics of all nodes（section 3.2）. Refine the parameter surfaces using Delaunay until the meshes meet the required quality（section 3.3）. Map the meshes onto surface physical spaces. The proposed algorithm can be used for both flat and curved surfaces. While curved surfaces require the use of a metric map, flat surfaces can use the identity matrix for the metric, provided an initial transformation of the three dimensional boundary nodes to the x-y plane is first accomplished. 3. Details of the Algorithm This section will define the metric and its uses. It will also describe the creation of the background mesh and the surface mesh that utilize the metric. 3.1 Definition of the metric Metric at a point: the metric tensor at a certain point P in 3D space that defines element size characteristics is a 3×3 matrix, M3p, which has the form [12] a b c 1 M== b d f e e e e e e T 3p 1 2 3 2 1 2 3 c f e 3 (1) where λi, ei(i=1, 2, 3) is the corresponding eigenvalue and eigenvector of Matrix M3p, and det(M3p)>0, λi>0. Metric specified on the surface: when parametric surface r (u, v) is meshed, the metric tensor of the surface itself and 3D points specialized by users need to be taken into account simultaneously. Then, we get the metric, Msp, at a certain point P on the surface, as follows τ1 τ ττν MMsp= 2 3 p () 1 2 ν 2 (2) rr ν = uv τ = r τ = r rr where 1 u , 2 v , uv and the notation of 2 indicates that only the first two lines and two columns of the matrix are considered. 3.2 Discretizing Boundary Curve Curve meshing is one of the main steps in the meshing process of planes, curved surface and volumes.